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MANTHAN: Journal of Commerce and Management
Volume 4, Issue 2, July-December 2017, pp. 13-28
doi: 10.17492/manthan.v4i02.11462
Modelling Volatility of BSE Realty Index using Conditional
Heteroscedasticity Models
Dhananjaya Kadanda* and Krishna Raj**
ABSTRACT
This study empirically examines the nature of volatility in BSE Realty Index using daily
closing price of BSE Realty Index for five and half years period from January 2011 to
June 2016. The study employed conditional heteroscedasticity models, both symmetric
and asymmetric, for the analysis. The study found that the volatility is persistent in the
Index return indicating the presence of volatility clustering in the series. Further, the
study reports the presence of asymmetric or leverage effect in the series as the leverage
coefficient γ is significant in EGARCH model indicating that negative shocks have
significant effect on volatility. However, the study did not find significant risk-return
trade-off in the series.
Keywords: Conditional volatility; Volatility clustering; GARCH model; Leverage effect.
1.0 Introduction
To gauge the market performance, and to convey the day to day market
movement to the investors, various indices are reported by stock exchanges all over the
world. Dow Jones Industrial Average (DJIA) of New York Stock Exchange (NYSE) is
one of the earliest stock market indices which appeared in May 26, 1896 which initially
reported the simple average of prices of 11 stocks on a particular day. Today, DJIA has
become one of the popularly cited indictors of market activity which reports weighted
average price of 30 widely traded stocks in NYSE. In India, S&P Sensex and S&P CNX
Nifty are the two benchmark indices which report the market movement of Bombay
Stock Exchange (BSE) and National Stock Exchange (NSE) respectively.
__________________
*Corresponding author; Doctoral scholar, Center for Economic Studies and Policy, Institute for
Social and Economic Change, Bengaluru, Karnataka, India. (Email id: dhananjayak@isec.ac.in)
** Professor, Center for Economic Studies and Policy, Institute for Social and Economic Change,
Bengaluru, Karnataka, India. (Email id: krishnaraj@isec.ac.in)
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MANTHAN: Journal of Commerce and Management, Volume 4, Issue 2, Jul-Dec 2017
While S&P Sensex is an index reported by BSE which is weighted average
prices of 30 major stock from various sectors, S&P CNX Nifty is an index reported by
NSE which is weighted average price 50 major stock from various sectors. Along with
these benchmark indices, both BSE and NSE also report various sectoral indices to track
the performance of these sectors thereby enabling more informed investment decisions.
Modelling and forecasting of the volatility of these indices has been one of the important
issues in the literature of financial economics. This is largely due to the importance of
volatility in financial market. Volatility, in simple, denotes the variability of an asset
price which is generally expressed in terms of variance or standard deviation of the
return of an asset (Gerety and Mulherin, 1991). Because this measure represents the
fluctuations in the asset return, it is also used as a simple measure of market risk. Greater
volatility is perceived by investors as greater risk in the asset which may threaten
investors’ assets and wealth (Karmakar, 2006). This will adversely affect the incentive to
save and to invest (Du and Wei, 2004). Higher and persistent variability in the asset
return over very short period of time may also result in investors’ loss of confidence in
the stock market which may eventually lead to withdrawal of investors from the market.
Further, increased volatility will result in higher cost of capital as investors now demand
premium for bearing greater risk in the asset (Edwards, 1988). Therefore, estimating and
forecasting volatility is critical in various financial activities such as portfolio choice and
risk management, derivative pricing and hedging, market making, market timing etc.
(Engle and Patton, 2001). This paper aims at examining nature of volatility in BSE
Realty return. Specifically, the study attempts understand whether the volatility in the
index is time varying, predictable, exhibit volatility clustering, and asymmetry behaviour
and risk-return trade off in the series.
2.0 Literature Review
Finance literature documented various features of financial time series such as
asset return which led the development of forecasting models. Firstly, as pointed out by
Schwert (1989) the variance of financial time series is non-constant or heteroskedastic.
Secondly, most of the financial time series exhibit the phenomenon of volatility
clustering wherein the large changes in asset price are followed by large changes
resulting the persistence of volatility too long in the future leading to connecting periods
of volatility and stability. Similarly, small changes are followed by small changes. This
results in volatility clustering or pooling in the return series. This phenomenon was first
documented by Mandelbrot (1963). Subsequently, it was also reported by Fama (1965)
who showed that the distribution of the asset returns tends to be fat-tailed. Thirdly, the
Modelling Volatility of BSE Realty Index using Conditional Heteroscedasticity Models
15
finance literature argued that the volatility caused by a negative shock to financial time
series would be more than a positive shock of the same amount. This is called leverage
effect in the case of equity returns as debt to equity ratio rises when the value of a stock
falls thereby increasing volatility of returns to equity holders (Nelson, 1991, Turner and
Weigel, 1992, and Brooks, 2008). However, conventional econometric models assume
constant variance or homoscedasticity and hence cannot model the series involving time
varying variance. Engle (1982) developed Autoregressive Conditional Heteroscedasticity
Models (ARCH) to deal with the time series which exhibits time varying conditional
variance in which the conditional variance is modelled to be dependent on the squares of
residuals. However, empirically ARCH model was found to be requiring a relatively
long lag in the conditional variance equation. Further, a fixed lag is imposed to
overcome the problem of negative variance parameters (Goudarzi, 2011).
To overcome these problems of ARCH model, Bollerslev (1986) developed
generalized the ARCH (GARCH) Model. In the GARCH model the conditional variance
depends not only on past squared errors but also on past conditional variance. To capture
the risk return relationship in the financial time series, Engle, Lilien and Robins (1987)
developed GARCH in Mean (GARCH-M) Under GARCH-M model the conditional
variance is included in the mean equation directly indicating that the return of an asset
may depend on the volatility. However, these GARCH models assume that there is
symmetric response of volatility to positive and negative shocks. This is because in
GARCH equations conditional variance is dependent upon value of the square of lagged
residuals which completely disregards the sign of that residuals (Turner and Weigel,
1992). However, the finance literature argued that a volatility caused by negative shock
to financial time series would be more than a positive shock of the same amount. This is
called leverage effect in the case of equity returns as debt to equity ratio rises when the
value of a stock falls thereby increasing volatility of returns to equity holders (Nelson,
1991, Turner and Weigel, 1992, and Brooks, 2008). To capture this asymmetric impact
of negative and positive shock on volatility Nelson (1991) developed Exponential
GARCH (EGARCH). Besides, allowing possible asymmetry in the series, EGARCH
also overcomes the problem of non-negativity constraints which is imposed on
symmetric GARCH parameters. Zokoian (1994) introduced alternative model, known as
Threshold GARCH, which replaced the quadratic specification of simple GARCH with a
piecewise linear function allowing for asymmetric effect of volatility. Glosten,
Jagannathan, and Runkle (1993) suggested another alternative asymmetry model, known
as GJR model, to allow asymmetric impact of shocks on volatility. The GJR model is a
simple extension of GARCH with an additional term added to account for possible
asymmetries. Subsequently, several other versions of GARCH such as Power GARCH,
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MANTHAN: Journal of Commerce and Management, Volume 4, Issue 2, Jul-Dec 2017
Component GARCH etc. were developed to better capture the characteristics of
volatility.
Empirically, many studies have attempted to model the volatility in Indian bench
mark indices. For instance, Karmakar (2006) estimated the volatility in Economics
Times Index and S&P CNX Nifty over the period 1961 to 2005 and found the evidence
of volatility clustering in the series. Karmakar (2006) also reported the presence of
asymmetry in volatility. Similarly, Karmakar (2007) examined the nature of volatility in
S&P CNX Nifty during July 1990 to December 2004 and found that volatility is time
varying, persistent and predictable. He also reported presence of volatility clustering in
the series. Similar results were also documented by Banumathy and Ramachandran
(2015), Kumar and Singh (2008), Joshi (2010), Banerjee and Sarkar (2006) in the case of
CNX Nifty. Goudarzi and Ramanarayanan (2010) studied the nature of volatility of the
BSE 500 and found that the volatility is persistent and exhibits clustering. Srinivasan and
Ibrahim (2010) examined the volatility in BSE Sensex and reported the presence of
leverage effect in the volatility. However, the risk-return tradeoff is not significantly
evident in Indian stock market (Banumathy and Ramachandran, 2015, Kumar and Singh,
2008, Karmakar, 2007). Most of the studies in India addressed the volatility of
benchmark induces like Sensex and Nifty. The volatility of sectoral indices has not been
understood. This study intends to partly bridge this research gap by studying the nature
of volatility of BSE Realty Index.
3.0 Data Sources and Research Methodology
The study is based on the secondary data that were collected from Bombay
Stock Exchange (BSE) Ltd, India. The daily closing prices of BSE Realty Index over the
period of five and half years from 1st January 2011 to 30th January 2016 were collected
and used for the analysis. Monthly data from January 2015 to June 2016 has been used to
understand the performance of Sensex, BSE Realty and companies of BSE Realty Index.
Daily data spanning from January 2010 to July 2016 has been used for computing
Granger Causality among BSE Realty companies and Sensex and BSE Realty. Further,
to understand the behaviour of US Realty, Sensex, and BSE Realty during Global
Financial Crisis 2008, daily data from September, 2008 to March 2009 has been used.
Data on US Realty has been collected from MSCI Inc.
3.1 Research methods
The study employed descriptive statistics, Granger causality tests and GARCH
techniques such as GARCH, GARCH-M, TGARCH and EGARCH to analyse the data.
Modelling Volatility of BSE Realty Index using Conditional Heteroscedasticity Models
17
Data has been analysed using E-views econometrics package. We ran different variants
of GARCH as the literature is not conclusive on the efficiency of any single model.
From the index, return series is generated as the first difference of log of daily
closing price which is as follows;
Equation (1) can be written as;
where logrt is the logarithm of daily return on index for time t, Pt is the closing price at
time t, and Pt−1 is the corresponding price in the period at time t − 1. Volatility is
measured as the standard deviation of the log return.
3.2 Symmetric GARCH models
As discussed in previous section, symmetric GARCH models assume that there
is symmetric response of volatility to both positive and negative shocks. The GACRH
model and GARCH-M model are used to study relation between return and volatility.
3.3 Generalized ARCH (GARCH) model
In the GARCH model the conditional variance (σ2t) depends on past squared
errors and past conditional variance (σ2t-s). So a simple GARCH (1, 1) model can be
written as;
Mean equation: rt = µ+ εt ……….(3)
Variance equation: σ2t =α0+ α1ε2t-1 + β1 σ2t -1 ……………(4)
3.4 GARCH-in-Mean (GARCH-M) model
Under GARCH-M model the conditional variance is included in the mean
equation directly indicating that the return of an asset may depend on the volatility.
A simple GARCH-M (1, 1) model can be shown as:
Mean equation: rt = µ+ γσ2t-1 + εt and ……….(5)
Variance equation: σ2t =α0+ α1ε2t-1 + β1 σ2t -1 ………………(6)
γ in equation (5) is interpreted as a risk premium (Brooks, 2008). If γ is positive
and statistically significant, then it indicates that return is positively related its volatility,
i.e. an increase in the conditional variance, leads to an increase in the mean return.
3.5 Asymmetric GARCH Models
To capture this asymmetric impact of negative and positive shock on volatility
Nelson (1991) developed EGARCH which can be specified as follows;
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MANTHAN: Journal of Commerce and Management, Volume 4, Issue 2, Jul-Dec 2017
This model allows for possible asymmetries in the series since if the relationship
between volatility and returns is negative, γ, will be negative. Another advantage of this
GARCH specification is that no artificial non-negativity constraints is required on the
model parameters. This is because since log (σ2t) is modelled, even if the parameters are
negative, σ2t will be positive (Brooks, 2008).
3.6 Threshold-GARCH
The T-GARCH model is a simple extension of GARCH with an additional term
added to account for possible asymmetries. The conditional variance is now given by,
where It−1 = 1 if εt-1< 0
= 0 otherwise
If γ > 0 in equation (8), there is leverage effect in the series.
4.0 BSE Realty Index: Some Stylized Facts
Real estate sector is one of the important sectors in the economy. Besides direct
contribution to GDP, it also generates demand in various other sectors like steel, cement,
bricks, paints, building materials, consumer durables etc. Construction sector, therefore,
has multiplier effect on the economy. Table 1 shows the growth and contribution of
construction sector. As shown in the table, during 2010 to 2015 the sector contributed an
average of 7.86 per cent to GDP and is expected to reach 13 per cent by 2028 (KPMG,
2014).
Table 1: Decadal Growth of GDP and Construction Sector
Decade
GDP (CAGR)
Construction sector
(CAGR)
Share of construction
sector in GDP
1950-59
3.7
5.9
5.40
1960-69
3.3
6.9
7.40
1970-79
3.4
3.1
7.35
1980-89
5.2
3.7
6.89
1990-99
6.1
4.8
6.64
2000-09
7.8
10.6
7.46
2010-15
6.85
5.43
7.86
Source: Authors’ construction
Modelling Volatility of BSE Realty Index using Conditional Heteroscedasticity Models
19
The sector is also one of the largest employers in the economy. Recognising the
contribution and the importance of real estate sector, BSE introduced BSE Realty Index
comprising of 11 scrips representing 95 per cent of real estate companies. The base year
is fixed as 2005, with base index value of 1,000. With the introduction of BSE Realty,
investors, both domestic and global, now have a benchmark index to track the movement
of real estate companies in the stock market which would help them in their investment
decision.
Figure 1 presents the movement of Sensex and Realty index during the last 10
years. It is clear from the figure that in the post crisis period, Sensex has expriend
upward trend on an average, except for 2011, Untill 2012, both Sensex and Realty Index
exhibited similar behavior. For instance, after a deep slump in 2008, both the indices
came strong in 2009. However, the recovery has been slow in the case of BSE Realty,
whereas Sensex reached its pre-crisis value in 2010, BSE Realty never regained its pre-
crisis value.
Figure 1: Movement of Sensex and BSE Realty Index
Source: Authors’ construction
In fact after 2012, BSE Realty has witnessed a downward trend with a marginal
recovery in 2016. On other hand, Sensex has witnessed a strong upward traend after
2012. As evident from the figure, Sensex and BSE Realty exhibited opposite trend in
their movement after 2012 indicating that movement in BSE Realty does not affect the
Sensex.This may be due to the fact that none of the companies comprising BSE Realty
figures in Sensex.
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MANTHAN: Journal of Commerce and Management, Volume 4, Issue 2, Jul-Dec 2017
4.1 Relationship between BSE Realty Index, Sensex and US Realty Index
In order to understand the relationship between US Realty Index and BSE Realty
Index and Sensex during US Subprime crisis 2008, we examined the direction of
causality between US Realty, BSE Realty Index and Sensex. As shown in the table 2,
there was unidirectional causality between US Realty and the causality runs from US
Realty. This signifies the deep financial integration of stock markets which becomes
more pronounced during economic or financial crisis leading to transmission of shocks
from one market to another.
Table 2: Causality Between BSE Realty, US Realty, and Sensex
Null Hypothesis
F-Statistic
Prob.
Inference
US Realty does not Granger Cause BSE Realty
4.44105
0.0359
Reject
BSE Realty does not Granger Cause US Realty
0.89119
0.3459
Accept
US Realty does not Granger Cause Sensex
15.0695
0.0001
Reject
Sensex does not Granger Cause US Realty
0.90031
0.3434
Accept
Sensex does not Granger Cause BSE Realty
7.88675
0.0053
Reject
BSE Realty does not Granger Cause Sensex
1.34363
0.2473
Accept
Source: Authors’ computation
4.2 Market performance of BSE Realty and Sensex companies
Performance of BSE Realty Index, Sensex and Index companies in the last 18
months has been presented in Table 3. As shown in Table 3, both BSE Realty and
Sensex recorded negative return in the last 18 months. Among BSE Realty companies,
Godrej, HDIL, Indiabulls, and Omaxe performed better, whereas NBCC, Oberoi,
Prestige, Sobha, and Unitech companies recorded negative return. While HDIL is the
best performer in the last 18 months with an average return of 41.4 per cent, NBCC
recorded the worst performance with -145.51 per cent. It is clear from the table that real
estate companies, on an average experienced poor performance in the stock market.
To understand whether BSE Realty companies move with market return, we ran
Granger causality between Sensex and these companies. Table 4 presents the results of
Granger causality test. As shown in the table, eight of eleven BSE Realty companies are
influenced by market return (Sensex). While no causality was found in the case of DLF,
Indiabulls, NBCC, market return was found to cause return of Godrej, HDIL, Oberoi,
Omaxe, Phoenix, Prestige, Sobha, and Unitech. Therefore, on an average the
performance of BSE Realty companies are influenced by the market return, whereas,
none of the BSE Realty companies were found to be causing market return.
Modelling Volatility of BSE Realty Index using Conditional Heteroscedasticity Models
21
Table 3: Return of BSE Realty Companies
Month
DLF
Godrej
HDIL
India bulls
NBCC
Oberoi
Omaxe
Prestige
Sobha
Unitech
Realty Index
Sensex
Jan-Mar
-15
0.06
-0.02
0.40
-0.05
0.16
0.02
0.08
0.12
-0.19
-0.02
0.07
0.02
Apr-
Jun-15
-0.30
-0.02
-0.10
-0.14
-0.06
-0.04
-0.01
-0.08
-0.11
-0.72
-0.16
-0.01
Jul-Sep-
15
0.16
0.21
0.37
-0.26
0.26
-0.05
0.14
-0.08
-0.85
-1.73
-0.20
-0.06
Oct-
Dec-15
-0.17
0.03
0.05
0.00
0.03
-0.02
0.01
-0.10
0.11
0.08
-0.04
0.00
Jan-Mar
-16
-0.01
-0.13
-0.06
-0.13
-0.05
-0.10
0.06
-0.12
-0.13
-0.30
-0.09
-0.03
Apr-
Jun-16
0.27
0.21
0.35
0.51
-1.60
0.12
0.10
0.07
0.18
0.26
0.22
0.06
Jan- 15-
Jun-16
(%)
0.43
35.35
41.40
30.17
-145.5
-2.58
24.63
-25.9
-39.1
-95.1
-1.44
-1.83
Source: Authors’ construction
Table 4: Granger Causality of BSE Realty Companies’ Return with Market Return
(Sensex)
Null Hypothesis
F-Statistic
Prob.
Inference
DLF return does not Granger Cause Sensex return
0.24
0.62
Accept
Sensex return does not Granger Cause DLF return
0.58
0.45
Accept
Godrej return does not Granger Cause Sensex return
1.11
0.29
Accept
Sensex return does not Granger Cause Godrej return
22.56
0.00
Reject
HDIL return does not Granger Cause Sensex return
0.84
0.36
Accept
Sensex return does not Granger Cause HDIL return
9.20
0.00
Reject
Indiabulls return does not Granger Cause Sensex return
1.18
0.28
Accept
Sensex return does not Granger Cause Indiabulls return
1.23
0.27
Accept
NBCC return does not Granger Cause Sensex return
0.26
0.61
Accept
Sensex return does not Granger Cause NBCC return
1.93
0.16
Accept
Oberoi return does not Granger Cause Sensex return
0.09
0.77
Accept
Sensex return does not Granger Cause Oberoi return
21.35
0.00
Reject
Omaxe return does not Granger Cause Sensex return
0.61
0.43
Accept
Sensex return does not Granger Cause Omaxe return
8.19
0.00
Reject
Pheonix return does not Granger Cause Sensex return
0.03
0.87
Accept
Sensex return does not Granger Cause Pheonix return
7.55
0.01
Reject
Prestige return does not Granger Cause Sensex return
0.80
0.37
Accept
Sensex return does not Granger Cause Prestige return
18.81
0.00
Reject
Sobha return does not Granger Cause Sensex return
0.00
0.96
Accept
Sensex return does not Granger Cause Sobha return
27.08
0.00
Reject
Unitech return does not Granger Cause Sensex return
1.46
0.23
Accept
Sensex return does not Granger Cause Unitech return
7.86
0.01
Reject
Source: Authors’ Computation
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MANTHAN: Journal of Commerce and Management, Volume 4, Issue 2, Jul-Dec 2017
5.0 Results and Discussions of Volatility Modelling
Table 5 summarises the descriptive statistics of the series. The average return of
the series is -0.000427 which indicates that the price has decreased over the period.
Table also shows that the series is positively skewed. Further, Kurtosis of the series is >
3 indicating that the series is fat tailed and does not follow normal distribution which is
confirmed by J-B test statistics. The volatility clustering of daily return of BSE Realty
return is given in Figure 2. Table 6 presents ADF and Philips-Perron test result for
stationarity of the variable. As depicted in the table calculated t statistic is more than
critical value at 5% level. So we reject null hypothesis that the series has unit root.
Therefore, the return series is stationary which enables us to proceed with the GARCH
model. However, one of the important conditions for running GARCH model on the
series is the presence of heteroscedasticity or ARCH effect in the residuals of the series.
To do this we employ ARCH LM test. Results of ARH LM test depicted in Table 6
confirm that there is ARCH effect in the return series hence this cannot be modelled with
the simple ARMA structure.
Table 5: Descriptive Statistics of Daily Return
Mean
-0.000427
Minimum
-0.115728
Kurtosis
4.350617
Median
0.000168
Std.dev.
0.021435
Jarque-Bera
114.03
Maximum
0.076884
Skewness
0.210036
N
1368
Source: Authors’ Estimation
Figure 2: Volatility Clustering of Daily Return of BSE Realty Index
Modelling Volatility of BSE Realty Index using Conditional Heteroscedasticity Models
23
Table 6: Result of Unit Root Test and ARCH-LM Test for Residuals
Value
ADF
PP
t-statistics
-32.89192
-32.8644
Prob.
0.0000
0.0000
Critical Value
1%
-3.434924
-3.434924
5%
-2.863447
-2.863447
10%
-2.567834
-2.567834
ARCH LM test statistics
10.65268
Prob.
0.0011
Source: Authors’ Estimation
Therefore, we employed four variants of GARCH technique, namely
Generalized ARCH model, The GARCH in Mean(GARCH-M), Exponential GARCH(E-
GARCH), and Threshold GARCH(T-GARCH) models to capture the volatility in the
return series.
5.1 Symmetric GARCH Models
Table 7 presents the results of GARCH (1, 1) and GARCH-M (1, 1) Model. The
parameters of GARCH model, i.e. constant (ω), ARCH (α) term and GARCH (β) term
are statistically significant at 1% level. The estimated value of β is substantially higher
than α which indicates that market has longer memory and shows that volatility is
persistent. Further, since the sum of α and β is close to unity (0.92) the shock will remain
for many future periods.
ARCH LM Test was conducted to know the presence of ARCH effect in
residuals. Test results show that there is no further ARCH effect remaining in the
residuals of the series which indicates that variance equation is properly specified. As
shown in the table, coefficient of conditional variance entered into mean equation is
positive, but it is statistically insignificant. Hence one can infer that there is no
significant impact of volatility on return the expected return, implying the absence of
risk-return tradeoff in the series.
To examine whether risk return tradeoff prevails in the return series, we ran
GARCH-M model. The parameters of variance equation in GARCH-M (1, 1) model are
statistically significant at 1% level. The sum of α and β is close to unity (0.92) which
implies that the shock will persist to many future periods. Further, ARCH LM test on
residuals of GARCH-M (1, 1) model shows that there is no ARCH effect implying that
the model is well specified.
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MANTHAN: Journal of Commerce and Management, Volume 4, Issue 2, Jul-Dec 2017
Table 7: Estimated Results of GARCH(1,1) and GARCH-M(1,1) Models
Coefficients
GARCH(1,1)
GARCH-M(1,1)
Mean
µ (constant)
-0.000331
-0.004847
Risk Premium λ
0.216924***
Variance
ω (constant)
0.000037*
0.0000377*
α(ARCH effect)
0.068486*
0.06896*
β(GARCH Effect)
0.851096*
0.849175*
α + β
0.919582
0.918135
Log likelihood
3339.607
3340.077
Akaike info. Criterion (AIC)
-4.876618
-4.875843
Schwarz info. Criterion(SIC)
-4.861352
-4.85676
ARCH-LM test heteroscedasticity
ARCH-LM test Statistics
0.736898
0.642347
Prob.
0.3908
0.423
Notes:*Significant at 1% level, ***Insignificant
Source: Authors’ estimation
5.2 Asymmetric GARCH Models
We employed EGARCH (1, 1) and T-GARCH (1, 1) to capture asymmetries in
the return series. Table 8 presents the results of EGARCH and T-GARCH model.
Table 8: Estimated Results of EGARCH (1, 1) and TGARCH (1, 1) Models
Coefficients
EGARCH(1,1)
TGARCH-M(1,1)
Mean
µ (constant)
-0.000302
-0.000409
Variance
ω (constant)
-2.436016*
0.0000382*
α(ARCH effect)
0.710758*
0.057733*
β(GARCH Effect)
-0.048571*
0.845613*
γ (leverage effect)
-0.048571**
0.026576***
α + β
0.662187
0.903346
Log likelihood
3340.224
3340.271
Akaike info. Criterion (AIC)
-4.876058
-4.876127
Schwarz info. Criterion(SIC)
-4.856975
-4.857044
ARCH-LM test heteroscedasticity
ARCH-LM test Statistics
0.0142
0.791731
Prob.
0.9052
0.3737
Notes: *Significant at 1%, **5%, ***Insignificant
Source: Authors’ estimation,
Modelling Volatility of BSE Realty Index using Conditional Heteroscedasticity Models
25
As shown in Table 8, constant (ω), ARCH (α) term and GARCH (β) term of
EGARCH (1, 1) model are statistically significant at 1% level. γ, the leverage effect is
negative and significant at 5% level indicating the presence of leverage effect in the
series. This implies that there is negative correlation between past return and future
return. Finally, ARCH LM test confirms that there is no further ARCH effect in the
return series which shows that EGARCH (1, 1) model is specified properly. We also ran
T-GACRH as an alternate model to test for possible asymmetry in volatility in the series.
However, the, the leverage effect is found to be statistically insignificant indicating that
both positive and negative shock will have same effect on the volatility.
5.3 Summary of findings
The following are the major findings of the above empirical analysis:
In GARCH (1, 1) model, the sum of the coefficient is (α+β) is closer to one which
indicates that volatility is persistent.
In GARCH-M(1,1), the risk premium coefficient is insignificant indicating the
absence of risk-return trade off in the series which means that the higher risk does not
necessarily lead grater return.
The leverage effect parameter γ of EGARCH model is significant at 5% level which
implies the presence of asymmetry effect in the series. The negative value of γ
indicates that there is negative relationship between past variance and future return.
In the symmetric estimate, GARCH(1,1) model found to be the best fit as it yielded
lowest AIC and SIC value and highest log likelihood value when compared to its
alternate model GARCH-M(1,1).
In the asymmetric estimate, T-GARCH seems to be an appropriate model as it
produced lowest AIC and SIC value and highest log likelihood value.
6.0 Conclusion
The study tested the volatility of BSE Realty Index with symmetric and
asymmetric GARCH models using the daily return of the index for the last five and half
years. Four different GACRH models have been employed to understand the volatility
clustering. After confirming the stationarity of the series, we employed GARCH (1, 1),
GARCH-M (1, 1), EGARCH (1, 1) and TGARCH (1, 1) models. In symmetry estimates,
we found that GARCH (1, 1) model is the best fit based on AIC, SIC and log likelihood
criteria. Similarly, in asymmetry estimates, EGARCH has been found to be the
appropriate model. In nutshell, we found that there is volatility persistence in the series
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MANTHAN: Journal of Commerce and Management, Volume 4, Issue 2, Jul-Dec 2017
resulting volatility clustering. We also found that there is leverage effect in T-GARCH
model as γ in the model turned out to be statistically significant. However, no risk return
trade-off in the series was found as γ parameter in GARCH-M (1, 1) model is
statistically insignificant.
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